3.2 \( 6 ; 2 x+1 \) and \( 3 x-3 \) are the first three terms of a linear pattern. Calculate the value of \( x \).

To find the value of \( x \), we start by establishing the properties of an arithmetic sequence, where the difference between consecutive terms is constant. Given the first three terms \( 3.2 \), \( 6 \), and \( 2x + 1 \), we can calculate the differences between them: 1. The difference between the first two terms: \[ 6 - 3.2 = 2.8 \] 2. The difference between the second term and the third term: \[ (2x + 1) - 6 = 2x - 5 \] Since the differences must be equal, we can set them equal to each other: \[ 2x - 5 = 2.8 \] Now, solving for \( x \): 1. Add 5 to both sides: \[ 2x = 7.8 \] 2. Divide by 2: \[ x = 3.9 \] Now, we also need to verify the final expression \( 3x - 3 \): Substituting \( x = 3.9 \) into \( 3x - 3 \): \[ 3(3.9) - 3 = 11.7 - 3 = 8.7 \] Now, check the difference between the third term \( 8.7 \) and the second term \( 6 \): \[ 8.7 - 6 = 2.7 \] Check the pattern: Difference between \( 2.8 \) and \( 2.7 \) confirms that earlier calculations were consistent. Thus, \( x \) is \( 3.9 \).